Krull dimension in power series rings

Abstract

Let R denote a commutative ring with identity. If there exists a chain P 0 ⊂ P 1 ⊂ ⋯ ⊂ P n {P_0} \subset {P_1} \subset \cdots \subset {P_n} of n + 1 n + 1 prime ideals of R, where P n ≠ R {P_n} \ne R , but no such chain of n + 2 n + 2 prime ideals, then we say that R has dimension n. The power series ring R [ [ X ] ] R[[X]] may have infinite dimension even though R has finite dimension.